Hello,
1) Product categories widget is the default WooCommerce widget. You can enable option Only show children of current category https://prnt.sc/widcns to show the subcategories list instead of the full list of categories. Filter by attributes appears if products have the attribute. For example:
Woman category https://xstore.8theme.com/demos/wedding/product-category/woman/ has size attribute https://prnt.sc/widelo
Bouquets https://xstore.8theme.com/demos/wedding/product-category/bouquets/ does not have the size but color attribute https://prnt.sc/widf42
The widgets area includes both filters https://prnt.sc/widfs9 but they appear depending on products that have these attributes.
2) Use the Stretch row option of the row settings to stretch the content width https://prnt.sc/widge8
3) Are you talking about Ajax showing of the products from the category? Add the Product Categories widget to shop sidebar (list view) and enable the Ajax Product Filters option at Theme Options > WooCommerce > Shop > Shop filters.
Regards